Precalculus Exponential Log Quiz: 10 Questions

Contents

Key takeaway: This precalculus exponential log quiz asks 10 multiple-choice questions on logarithm forms, graphs, laws, equations and models. A few items show a curve, so you must read the graph before you choose. The topics match F-LE.A.4. Every explanation points to the usual wrong turn, such as splitting the log of a sum. Plan for about 15 minutes.

Ten multiple-choice questions on logarithm forms, log laws, exponential and logarithmic equations, logistic models and logarithmic scales. Pick one answer for each question, then read the explanations to learn what each wrong choice gets wrong.

PrecalculusCommon Core: F-LE.A.4 · F-BF.B.5 · F-IF.C.8b

🎯 Exponential and Logarithmic Functions and Models – Precalculus math quiz

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Question 1

Which equation is the logarithmic form of \(4^{3/2}=8\)?

A \(\log_{4} 8=\frac{3}{2}\)

B \(\log_{8} 4=\frac{3}{2}\)

C \(\log_{3/2} 8=4\)

D \(\log_{4}\frac{3}{2}=8\)

Question 2

A curve with three marked points.

Which function has the graph shown?

A \(y=2^{x}\)

B \(y=\log_{1/2} x\)

C \(y=\log_{2} x\)

D \(y=\log_{2} x+1\)

Question 3

Which expression equals \(\ln 3+\ln 4\)?

A \(\ln 12\)

B \(\ln 7\)

C \(\ln\frac{3}{4}\)

D \(\ln 81\)

Question 4

Which expression equals \(2\log_{b} x-\log_{b} y\)?

A \(\log_{b}\frac{2x}{y}\)

B \(\log_{b}\frac{x^{2}}{y}\)

C \(\log_{b}(x^{2}-y)\)

D \(\log_{b}(x^{2}y)\)

Question 5

What is the solution of \(5^{x}=40\), rounded to two decimal places?

A \(x=\ln 8\approx 2.08\)

B \(x=\frac{\ln 40}{5}\approx 0.74\)

C \(x=5\ln 40\approx 18.44\)

D \(x=\frac{\ln 40}{\ln 5}\approx 2.29\)

Question 6

Solve \(\ln x+\ln (x-3)=\ln 4\).

A \(x=4 only\)

B \(x=4 and x=-1\)

C \(x=-1 only\)

D \(x=3.5\)

Question 7

A population over time, with a dashed horizontal line.

What is the carrying capacity shown by the graph?

A 300

B 60

C 1,200

D 600

Question 8

Money grows continuously at 3% per year. What is the doubling time, to the nearest tenth of a year?

A 33.3 years

B 23.1 years

C 0.04 years

D 2.31 years

Question 9

The graph of f(x) = 3^x.

The graph of \(f(x)=3^{x}\) contains the point shown. Which point must be on the graph of \(f^{-1}\)?

A \((2, 9)\)

B \((9, 2)\)

C \((-2, 9)\)

D \(\left(\frac{1}{9}, 2\right)\)

Question 10

A sound at 80 dB is how many times as intense as a sound at 60 dB?

A 20 times

B 10 times

C 100 times

D 1.33 times

🎉 Quiz complete!

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Remember:

log_b x = y means b^y = x, with b > 0, b not equal to 1 and x > 0.
Product, quotient and power laws: log(MN) = log M + log N, log(M/N) = log M – log N, log(M^k) = k log M.
Solve b^t = c with t = ln c / ln b, and always check log equations for extraneous answers.
Continuous growth is A = Pe^(rt); a logistic model P = K/(1 + Ae^(-rt)) levels off at K.
Log scales (pH, decibels, magnitude): a difference of d units means a ratio of 10^d.

Which questions test forms and laws?

Rewriting \(4^{3/2}=8\) as \(\log_{4}8=\frac{3}{2}\) opens the quiz. Later, \(\ln 3+\ln 4\) combines into \(\ln 12\) in question 3. Then \(2\log_{b}x-\log_{b}y\) becomes \(\log_{b}\frac{x^{2}}{y}\) in question 4. A tempting wrong choice moves the 2 incorrectly, so check where each exponent lands.

How do the equation questions work?

Solving \(5^{x}=40\) gives \(x\approx 2.29\) in question 5. In question 6, the equation \(x(x-3)=4\) offers both 4 and \(-1\), but only 4 survives the domain test. Meanwhile, question 9 relies on the inverse idea, because a curve for \(3^{x}\) holding \((2,9)\) forces the swapped pair \((9,2)\) onto the log curve.

What about the model questions?

A dashed line marks a carrying capacity of 600 in question 7. For continuous growth at 3%, the doubling time is 23.1 years. Finally, a decibel gap of 20 in question 10 means 100 times the intensity. Read the scale first, because each 10 dB step multiplies intensity by 10.

This quiz rewards care with signs and domains, so check every answer once.

What comes next for you?

Count your hits, then reread the lesson section that matches each miss. Redo the missed question without choices, writing the full solution. If you finish with nine or ten, move to the test.